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How are you feeling about Christmas lunch with the relatives? Worried you'll regress to your teenage years and all start pushing each other's buttons? The answer to this, as to all other questions, comes from maths: it pays to be nice!

We were lucky enough to travel to Cambridge, MA, to meet Nowak and his colleagues and find out more about his research. You can read our interview with him and other stories about the maths of altruism in our package and ebook.

We have a Mega Menger! Thanks to the tireless effort of students, the Centre for Mathematical Sciences in Cambridge (home of Plus) now has a piece of the largest distributed fractal model in the world. It's a model of a Menger sponge made out of over 48,000 business cards that are held together without any glue or sticky tape. Sets of six cards were folded up to form little cubes, a total of 8,000 of them, which together form an approximation of the sponge. The final result, which now adorns our common room, weighs over 90kg and is 1.5 metres tall.

The sponge model build was started off at an public event back in October, using thousands of little cubes that had been pre-built by students from twenty Cambridgeshire schools, coordinated by the Further Mathematics Support Programme. Enthusiastic members of the public and university students then finished it off, at one point working by the light of a single bulb when the building had shut down for the night.
The effort is part of the
MegaMenger project, which aims to build fractal models in multiple sites worldwide. If you would like to see our Mega Menger model, come to the Maths Faculty event at the 2015 Cambridge Science Festival, Saturday 21 March 2015, 12 noon — 4pm. You can find out more about fractals and the Menger sponge here.

"Imagine the scene of a violent murder where the victim has been bludgeoned to death."

This isn't the kind of sentence we were expecting to deal with when we agreed to help edit a book to celebrate the fiftieth anniversary of the Institute of Mathematics and its Applications (IMA). But we did. The book contains fifty articles about maths in all its shapes and guises — the gruelling sentence comes from one that explores how analysing the shape of blood stains can tell you how someone was killed.

It's not all that violent though: there are articles about champagne, Sherlock Holmes, space travel, gambling, sports, and much more besides, written by some of the best authors in maths and science. The foreword is by Dara Ó Briain and if you get tired of reading, you can look at the selection of fifty beautiful mathematical images that adorn the book. We have featured some of them as our images of the week.

Happy birthday IMA!

This image by Tim Jones, showing the ceiling of the Sagrada Familia basilica in Barcelona, is one of the images featured in the book.

On this 13th day of our Christmas Advent we thought you might like to think about your true love, and for that you need your maths.

You might find it surprising but mathematics provides some of the most profound metaphors for love and human relationships. From converging infinite sequences describing the attraction and force of love in A disappearing number to Elizabeth Barrett Browning's "How do I love thee, let me count the ways…", the rich language in mathematics provides just an accurate and evocative description of human emotions as it rigorously defines mathematical concepts.

The number 12 is very flexible. You can write it as 2 x 6 or as 3 x 4. Or, even better, as 2 x 2 x 3. Which brings us to those very special numbers called the primes: those numbers that are not divisible by any other number apart from 1 and themselves. Every number can be decomposed into a product of primes, for example 12 = 2 x 2 x 3. The primes are the atoms of number theory, they are also connected to one of the hardest open problems in maths, and even to the weird world of quantum physics.

As is tradition we start with a fact about this Carnival's number: 117. It's divisible by 3. That might not be so exciting to everyone, but to someone such as myself, as a mathematician who is notoriously bad at numbers and arithmetic (my friends think it's hilarious to ask me to divide the bill at a restaurant), being able to tell this quickly about any integer gives me a feeling of vast power over these numbers. You can tell if a number is divisible by 3 if the sum of the number's digits is divisible by 3: in our case, 1+1+7=9. Neat!

This trick gives me a great sense of power over the integers as it allows me to rule out vast swathes of them (well, a third) as not being prime, at a glance. Combine that with our instant knowledge that any even numbers, or those ending in 5 or 0 aren't prime, and you've really got a good of chance of being able to tell if any random number someone tells you is prime. There's other neat division rules too, for 7 and 11 and so on. But the really interesting thing is why they work… Can you explain why the division rule for three works? Give us your best explanations in the comments below!

Now to the Carnival. We had many great submissions, thank you to everyone who sent one in. The one that had me hooked for the longest was the wonderful Parable of the Polygons by Vi Hart and Nicky Case, submitted by Katie. This brilliant interactive post explores the maths of diversity based on the work of Nobel Prize-winning game theorist Thomas Schelling.

I also really enjoyed Kevin Houston's fascinating and detailed article The Beatles' magical mystery chord, delving into the mathematics and music behind the opening chord to a hard days night. It's made me want to catch the next talk Kevin gives on the maths of autotune!

A very mathematical snowflake

For a real treat, why not spend a long lunch hour watching Terry Tao's talk The Cosmic Distance Ladder. This forms part of one of Mike Lawler's blog post about the lecture, where he discuses how the radius of the Earth and the radius of the Moon's orbit were measured, and how Tao's lecture was an incredible opportunity to show some fun and exciting maths to children. Lawler said in his submission: "Amazingly this lecture by Tao on youtube has fewer than 2,000 views right now. I wish that more people knew about it (kids especially) as it is such an amazing opportunity to learn from one of the greatest mathematicians living today." We heartily agree! (And you can find out more on Tao's work here.)

I, and any other people madly manufacturing paper snowflakes for decorating school halls and homes, must thank Mr Reid for his excellent post A public service announcement regarding paper snowflakes. Really, if you're in the middle of cutting a million and one of those suckers out you need to read this, it will make it all much better!

And here are some more treats to while away your paper-snowflake-cutting-time…

That's it for this Carnival, we hope you enjoyed the show! You can find the last Carnival at CavMaths and the details for all past and future Carnivals, as well as how to be involved at Aperiodical. Don't forget to submit ideas for January's Carnival, hosted by Andrew at AndrewT.net.

(Hint for the division rule: A number with digits abcd=ax1000+bx100+cx10+d = ax(1+999)+bx(1+99)+c(1+9)+d…)